Symmetrization of Martingale Posterior Distributions

Music City Center 

The martingale posterior framework, recently proposed by Fong et al., is based on a sequence of one step ahead predictive distributions. It leads to computationally efficient inference in parametric and nonparametric settings. The predictive distributions implicitly provide a joint model for an infinite sequence of data. The observed data, arbitrarily considered to be Y1 through Yn, form the beginning of the sequence and the tail of the sequence is missing. Filling in the missing data allows one to summarize Y1 through Y∞ (or through YN, N large in practice). A typical summary, such as the mean, is regarded as a parameter.

In cases where the martingale posterior model does not match a de Finetti model, the joint distribution over the Y's is not exchangeable, and so the indices { 1, ..., n } of the data affect the analysis. We investigate methods of symmetrizing inference in these models. In some settings, re-indexing the observed data to { a1 < ... < an } and sending a1 to infinity is analytically tractable, and we recover classical Bayesian models with known priors. Additionally, we investigate the effect of nesting the nonexchangeable models within exchangeable models.

Martingale posterior

Bayesian nonpamametrics

Nonexchangeable models

Predictive inference

Bayesian uncertainty 

Section on Bayesian Statistical Science